Re: I need help.

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Mallinckrodt's Maple calculation:

> As has been pointed out, the conductive paper imposes boundary
> conditions that may be dealt with via an image charge technique.
> I have asked Maple to do some calculations to help me visualize
> the results.  ...

> ...  It is approximated here by a calculation based on the image
> charge technique.  I have superimposed the potential contributions
> from five other mirror dipoles that abut this region ....

John, how did you justify truncating the contributions from the
lattice of images after including only 5 image dipoles?  I believe
the Ewald sum of potentials over the lattice of images converges
quite slowly with distance from the original unit cell of the actual
paper.  In fact the Ewald sum is only conditionally convergent.  If
all the positive charges are summed first and then the negative ones
are subtracted from them the result is a difference of infinities.
If the image contributions are summed as charge neutral dipoles
the sum is still conditionally convergent.  The sum is only
significantly convergent when the dipoles are summed in nearest
neighbor pairs of dipoles whose overall dipole moment vanishes for
each dipole pair.  This requires a sum over a lattice of clusters
of (at least) 4 point charges whose net charge and dipole moment
vanish for each cluster.  Even in this case I don't think the rate
of convergence is very impressive.  It doesn't seem to me that adding
up 5 image dipoles plus the original dipole in the paper's unit cell
is going far enough out to get a decent approximation for the total
potential.  Did you maybe approximate the contribution from the more
distant dipole pairs (actually these pairs have a quadrupole moment
as their lowest order nonvanishing multipole moment) by some sort of
continuum approximation scheme?

David Bowman
David_Bowman@georgetowncollege.edu